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The projection technique of Sanchez-Portal et al. [48] have been implemented. The eigenstates tex2html_wrap_inline2684 obtained from PW calculations when sampling at a given wavevector tex2html_wrap_inline2533 are projected onto Bloch functions formed from a localised basis set tex2html_wrap_inline2688 . In this case, the natural choice of basis set is the set of atomic pseudo-orbitals generated from the pseudopotentials used in the calculation. However, care must be taken when performing the projection because this basis set is neither orthonormal nor complete.

The overlap matrix of the localised basis set, tex2html_wrap_inline2690 , is defined :


These overlaps are computed in reciprocal space, as the imposition of periodic boundary conditions implies that the orbitals need only be evaluated on a discrete reciprocal space grid. The overlaps between orbitals on different atomic sites can be calculated on the same grid with the application of a phase factor. The representations of the atomic pseudo-orbitals on the discrete reciprocal space grid may be calculated by Fourier transformation of the real space wavefunctions generated during construction of the pseudopotentials, or can be generated using the CASTEP code.

The overlaps between the plane wave states and the basis functions,


are also calculated in reciprocal space, as this is the natural representation of the plane wave states.

The quality of the projection may be assessed by the calculation of a spilling parameter,


where tex2html_wrap_inline2692 is the number of PW eigenstates, tex2html_wrap_inline2694 are the weights associated with the calculated tex2html_wrap_inline2533 points in the Brillouin zone and tex2html_wrap_inline2698 is the projection operator of Bloch functions with wave vector tex2html_wrap_inline2533 generated by the atomic basis.


where tex2html_wrap_inline2702 are the duals of the atomic basis states such that


The spilling parameter varies between one, in the case that the atomic basis set is orthogonal to the PW eigenstates and zero, when the projected wavefunctions perfectly represent the PW states. Unlike the procedure of Sanchez-Portal et al. [48], the basis functions are not scaled in order to optimise tex2html_wrap_inline2704 . This gives spilling parameters in the region of tex2html_wrap_inline2706 which are believed to be adequate for the purposes of this work.

The density operator may be defined,


where tex2html_wrap_inline2708 are the occupancies of the PW states, tex2html_wrap_inline2710 are the projected eigenstates, tex2html_wrap_inline2712 , and tex2html_wrap_inline2714 are the duals of these states. From this, the density matrix for the atomic basis set may be calculated as follows:


tex2html_wrap_inline2716 may be calculated in a time proportional to the cube of the system size, the same order of scaling as the original PW calculations, but the total time is smaller as there is no need to carry out iterations to achieve self-consistency.

The density matrix tex2html_wrap_inline2718 and the overlap matrix tex2html_wrap_inline2690 are sufficient to perform population analysis of the electronic distribution. In Mulliken analysis [43] the charge associated with a given atom A is given by


and the overlap population between two atoms A and B is


In Löwdin analysis [44] these quantities are given by




In Equations (gif) and (gif) the matrix tex2html_wrap_inline2728 is calculated as a Taylor expansion of tex2html_wrap_inline2730 where tex2html_wrap_inline2732 .

Details of the implementation of these methods may be found in Appendix gif.

next up previous contents
Next: Application to Simple Molecules Up: Population Analysis Previous: Introduction

Matthew Segall
Wed Sep 24 12:24:18 BST 1997